The ability to perform spatial tasks is crucial for everyday life and of great importance to cognitive agents such as humans, animals, and autonomous robots. A common artificial intelligence approach to accomplish spatial tasks is to represent spatial configurations and tasks in form of detailed knowledge about various aspects of space and time. Suitable algorithms then use the representations to compute solutions to spatial problems. In comparison, natural embodied and situated agents often solve spatial tasks without detailed knowledge about geometric, topological, or mechanical laws; they directly relate actions to effects that are due to spatio-temporal affordances in their bodies and environments. Accordingly, we propose a paradigm that makes the spatio-temporal substrate an integral part of the engine that drives spatial problem solving. We argue that spatial and temporal structures in body and environment can substantially support (and even replace) reasoning effort in computational processes: physical manipulation and perception in spatial environments substitute formal computation. While the approach is well known – for example, we employ diagrams as spatial substrate for geometric problem solving and maps for wayfinding – the underlying principle has not been systematically investigated or formally analyzed as a paradigm of cognitive processing. Topology, distance, and orientation constraints are all integrated and interdependent in truly 2- or 3-dimensional space. Exploiting this fact may not only help overcome the need for acquiring detailed knowledge about the interrelationships between different aspects of space; it also can point to a way of avoiding exploding computational complexity that occurs when we deal with these aspects of space in complex real-world scenarios. Our approach employs affordance-based object-level problem solving to complement knowledge-level formal approaches. We will assess strengths and weaknesses of the new cognitive systems paradigm.